How to measure the “well-roundedness” of SE contributors?

Stack Exchange, as we all know it, is a collection of Q&A sites with diversified topics. Assuming that each site is independent from each other, given the stats a user has, how to compute his "well-roundedness" as compared to the next guy? What is the statistical tool I should employ?

To be honest, I don't quite know how to mathematically define the "well-roundness", but it must have the following characteristics:

1. All things being equal, the more rep a user has, the more well-rounded he is
2. All things being equal, the more sites a user participates in, the more well-rounded he is.
3. Whether answer or question doesn't affect the well-roundness
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 How would you define "well-roundedness" in the SE context, first of all? – J. M. May 12 '11 at 13:34 @J.M, I don't know, that's why I would hope that the community would help to flesh out the concepts a little bit more. – Graviton May 12 '11 at 13:53 At the risk of sounding like one of the knee-jerk PC police - Let's be inclusive of the women who visit our site. @Graviton, All of your references are to males. – rolando2 May 15 '11 at 13:29

You need to account for similarity between the sites, as well. Someone who participates on StackOverflow and Seasoned Advice is more well-rounded than someone who participates on SO and CrossValidated, who is in turn (I would argue) more well-rounded than someone who participates in SO and Programmers. There are undoubtedly many ways to do that, but you could check overlapping registration to just get a feel for it.

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@Matt Parker, good points - If rep is part of the metric, I also think you need to look at how the rep was gained. A person that gained their rep from small gains on a lot of questions/answers would likely be more well rounded than a person who gained the same rep on a single question that garnered a ton of up-votes. – KennyPeanuts May 12 '11 at 18:49
@Kenny That's a good point and I think the way you've stated it is correct - but I feel like many of the most highly-voted questions and answers are often least indicative of expertise, whereas good answers to really gritty technical questions often have very low votes. – Matt Parker May 12 '11 at 19:14
Check out whuber's answers, for example. His answers are uniformly excellent, so what differentiates between his highly-rated answers and his 1-voters? When I look at the former, I see superb answers to questions that I understand; at the latter, superb answers to questions I don't even begin to grok. So low-voted answers can actually be indicative of deep specialization (but note: if you're curious about what a well-rounded SE user looks like, this is it). – Matt Parker May 12 '11 at 19:16
In fact, @Graviton, it might be wise to seek out multi-site users that you'd consider to be well-rounded and see how they perform in your various metrics. – Matt Parker May 12 '11 at 19:18
@KennyPeanuts, to complement @Matt's point above, there seem to also be an appreciable number of users on SE sites that respond to a tremendous number of questions, yet yielding few upvotes. That's not usually indicative of expertise or well-roundedness, either. (Though, it is one of the easiest strategies to garner "reputation"). – cardinal May 15 '11 at 14:09

EXAMPLE: say there are three sites, and we want to compare the well-roundedness of the Users A, B, C. We write the reputations of the users across the three sites in vector form:

User A: [23, 23, 0]

User B: [15, 15, 0]

User C: [10, 10, 10]

We would consider A more well-rounded than B (their reputations are both spread out evenly across two sites, but A has more total reputation). Also, we would consider C more well-rounded than B (they have the same total reputation, but C has an even spread across more sites.) It is undecided whether A should be considered more well-rounded than C, or vice-versa.

Let $x_A$, $x_B$, $x_C$ be the above reputation vectors respectively.

We want to measure the "well-roundedness" of a user by a function of their reputation vector $f(x)$. By the above, we would want our function $f$ to satisfy $f(x_A) > f(x_B)$, and $f(x_C) > f(x_B)$.

Any $f(x)$ that is concave and increasing will do the trick.

Two common examples of convex functions are the 'fractional norm'

$$f([x_1,...,x_m]) = \sum_i x_i^p$$

for $0 < p < 1$.

Taking $p = 1/2$, we calculate

$$f(x_A) = 2\sqrt{23} \approx 9.6$$ $$f(x_B) = 2\sqrt{15} \approx 7.7$$ $$f(x_C) = 3\sqrt{10} \approx 9.5$$

According to the $1/2$-norm, User A would be considered the most well-rounded of the three, by a narrow margin over User C.

Another choice for $f$ is the (scaled) Shannon entropy

$$f([x_1,...,x_m]) = -\sum_i x_i \log(x_i/c).$$

where $c = \sum_i x_i$.

If we take $f$ to be the scaled Shannon entropy, then we calculate

$$f(x_A) = 46 \log(2) \approx 31.9$$ $$f(x_B) = 30 \log(2) \approx 20.8$$ $$f(x_C) = 30 \log(3) \approx 33.0$$

Measured according the scaled Shannon entropy, then, we would say C is the most well-rounded of the three, and A the second most well-rounded.

EDIT: I originally said the function $f(x)$ had to be convex; the opposite is true.

EDIT2: Added an example in light of whuber's comment.

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 Re the edit: It's hard to know whether you're correct or not because there's nothing in your reply that describes what role $f$ plays: you never use it nor do you say how it would be applied in this problem. – whuber♦ May 15 '11 at 14:04 (+1) Thanks for the edit: that makes the idea much clearer. You might want to make the example match the counts, though: it seems the example concerns three users and three sites rather than four of each. BTW, how do you determine which sites to use? Nobody is active on all, or even a majority, of the sites. This is a problem because some people get 1 rep point to start with and others get 101. If we insist $f$ be constant in the interval $[1,101]$ to deal with this, then it's impossible for $f$ to be concave and increasing for any larger values! – whuber♦ May 15 '11 at 19:19

If you define 'well-roundedness' as 'contributing to many different Stack Exchange Sites,' I would compute some metric of contribution per site. You could use total posts, or average posts per day, or perhaps reputation. Then look at the distribution of this metric across all sites, and compute its skewness in some way that makes sense.

In other words, a 'well-rounded' person would be one who contributes to many different sites, while a 'not well-rounded' person would be one who primarily contributes to one site. You could further improve this by scaling your metric with a user's total across all sites. i.e. someone who's contributed a lot to many different sites should be considered more well-rounded than someone who's contributed nothing to any of the sites. A person who's never used SE isn't very well rounded!

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This is a really, really interesting question (indeed I'm somewhat in love with the idea of modelling the stack exchange sites in general).

On the issue of well-roundedness, one way to assess this would be through the tags that particular users tend to answer, and their distribution across sites. Examples may make this clearer.

I am a member on TeX, StackOverflow, CrossValidated and AskUbuntu. Now, I really only contribute to here and StackOverflow, and only about R on Stackoverflow. So,to define well roundedness I would look at a) the amount of tags which two sites have in common (to define similarity across sites) and the extent to which a user answers questions on sites which have little or no tags in common.

If, for instance, someone contributes to Python tags on StackOverflow and cooking, that person is more well-rounded than someone who is answering questions statistical software questions (for instance) on Overflow and stats questions here.

I hope this is somewhat helpful.

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(+1) Someone who is contributing to Python tags on both SO and cooking has some exotic tastes in food :-) I hear it tastes like chicken. – whuber May 15 '11 at 14:01